#6963038

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1172 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2)(3,11)(4,5)(6,9)(7,10)(8,14)(12,15)(13,16)(19,25)(22,26)(27,28)(30,31)(33,34,39,41,37,43,38,44)(35,45,53,57,47,40,48,55)(36,49,54,60,51,42,52,58)(46,56,62,63,50,59,61,64);(1,3)(2,8)(4,12)(5,13)(6,15)(7,14)(9,16)(10,11)(17,18)(19,26)(20,23)(21,24)(22,25)(27,31)(28,30)(29,32)(33,35)(34,40)(36,50)(37,47)(38,48)(39,53)(41,55)(42,59)(43,45)(44,57)(46,51)(49,56)(52,62)(54,61)(58,64)(60,63);(1,4,7,9)(2,5,10,6)(3,12,14,16)(8,13,11,15)(17,19,21,25)(18,22,24,26)(20,28,29,27)(23,31,32,30)(34,41)(36,51)(38,39)(40,55)(42,60)(43,44)(45,57)(46,50)(48,53)(49,58)(56,64)(59,63);(1,5,7,6)(2,9,10,4)(3,13,14,15)(8,16,11,12)(17,20,21,29)(18,23,24,32)(19,27,25,28)(22,30,26,31)(33,36,37,51)(34,42,43,49)(35,46,47,50)(38,52,39,54)(40,56,45,59)(41,58,44,60)(48,61,53,62)(55,63,57,64);(1,6,7,5)(2,9,10,4)(3,13,14,15)(8,12,11,16)(18,24)(22,26)(23,32)(30,31)(35,47)(36,51)(40,45)(42,49)(48,53)(52,54)(55,57)(58,60)

Polycyclic presentation:

2^9|a^2=ghi,c^2=fhi,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6870234);